|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/01.20.21.4303.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Integrate[(A Sinh[z]^2 + B Sinh[2 z] + C Cosh[z]^2)/
(a Sinh[z]^2 + b Sinh[2 z] + c Cosh[z]^2), z] ==
((-4 b B + a (A + C) + c (A + C)) z +
((-2 a b B + 2 b B c + A (2 b^2 - c (a + c)) + a^2 C - 2 b^2 C + a c C)/
Sqrt[-b^2 + a c]) ArcTan[(b + a Tanh[z])/Sqrt[-b^2 + a c]] +
((-A) b + a B + B c - b C) Log[-a + c + (a + c) Cosh[2 z] +
2 b Sinh[2 z]])/((a - 2 b + c) (a + 2 b + c))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["A", " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "2"]]], "+", RowBox[List["B", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["C", " ", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]]]]]], RowBox[List[RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "2"]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b", " ", "B"]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["A", "+", "C"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["A", "+", "C"]], ")"]]]]]], ")"]], " ", "z"]], "+", RowBox[List[FractionBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "b", " ", "B"]], "+", RowBox[List["2", " ", "b", " ", "B", " ", "c"]], "+", RowBox[List["A", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List["a", "+", "c"]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "C"]], "-", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", "C"]], "+", RowBox[List["a", " ", "c", " ", "C"]]]], ")"]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", RowBox[List["a", " ", "c"]]]]]], RowBox[List["ArcTan", "[", FractionBox[RowBox[List["b", "+", RowBox[List["a", " ", RowBox[List["Tanh", "[", "z", "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", RowBox[List["a", " ", "c"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "A"]], " ", "b"]], "+", RowBox[List["a", " ", "B"]], "+", RowBox[List["B", " ", "c"]], "-", RowBox[List["b", " ", "C"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "c"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]]]], ")"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mfrac> <mrow> <mrow> <mi> C </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> A </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> B </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> B </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> A </mi> <mo> + </mo> <mi> C </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> A </mi> <mo> + </mo> <mi> C </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> C </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> B </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> C </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> B </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> A </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> C </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> A </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mi> C </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> B </mi> </mrow> <mo> + </mo> <mrow> <mi> B </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> C </ci> <apply> <power /> <apply> <cosh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> A </ci> <apply> <power /> <apply> <sinh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> B </ci> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <sinh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <cosh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> <ci> c </ci> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <ci> b </ci> <ci> B </ci> </apply> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> A </ci> <ci> C </ci> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <plus /> <ci> A </ci> <ci> C </ci> </apply> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> C </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> B </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <ci> c </ci> <ci> C </ci> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> B </ci> <ci> c </ci> </apply> <apply> <times /> <ci> A </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <apply> <plus /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> C </ci> </apply> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <apply> <tanh /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> A </ci> </apply> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> C </ci> <ci> b </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> B </ci> </apply> <apply> <times /> <ci> B </ci> <ci> c </ci> </apply> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> c </ci> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> c </ci> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["A_", " ", SuperscriptBox[RowBox[List["Sinh", "[", "z_", "]"]], "2"]]], "+", RowBox[List["B_", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z_"]], "]"]]]], "+", RowBox[List["C", " ", SuperscriptBox[RowBox[List["Cosh", "[", "z_", "]"]], "2"]]]]], RowBox[List[RowBox[List["a_", " ", SuperscriptBox[RowBox[List["Sinh", "[", "z_", "]"]], "2"]]], "+", RowBox[List["b_", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z_"]], "]"]]]], "+", RowBox[List["c_", " ", SuperscriptBox[RowBox[List["Cosh", "[", "z_", "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b", " ", "B"]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["A", "+", "C"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["A", "+", "C"]], ")"]]]]]], ")"]], " ", "z"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "b", " ", "B"]], "+", RowBox[List["2", " ", "b", " ", "B", " ", "c"]], "+", RowBox[List["A", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List["a", "+", "c"]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "C"]], "-", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", "C"]], "+", RowBox[List["a", " ", "c", " ", "C"]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["b", "+", RowBox[List["a", " ", RowBox[List["Tanh", "[", "z", "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", RowBox[List["a", " ", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", RowBox[List["a", " ", "c"]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "A"]], " ", "b"]], "+", RowBox[List["a", " ", "B"]], "+", RowBox[List["B", " ", "c"]], "-", RowBox[List["b", " ", "C"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "c"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], "]"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|